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A005995 Alkane (or paraffin) numbers l(8,n).
(Formerly M2916)
7
1, 3, 12, 28, 66, 126, 236, 396, 651, 1001, 1512, 2184, 3108, 4284, 5832, 7752, 10197, 13167, 16852, 21252, 26598, 32890, 40404, 49140, 59423, 71253, 85008, 100688, 118728, 139128, 162384, 188496, 218025, 250971, 287964, 329004, 374794, 425334, 481404, 543004 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

From M. F. Hasler, May 01 2009: (Start)

Also, number of 5-element subsets of {1,...,n+5} whose elements sum to an odd integer, i.e. column 5 of A159916.

A linear recurrent sequence with constant coefficients and characteristic polynomial x^9 - 3*x^8 + 8*x^6 - 6*x^5 - 6*x^4 + 8*x^3 - 3*x + 1. (End)

Equals (1/2)*((1, 6, 21, 56, 126, 252,...) + (1, 0, 3, 0, 6, 0, 10,...)).

Equals row sums of triangle A160770.

F(1,5,n) is the number of bracelets with 1 blue, 5 identical red and n identical black beads. If F(1,5,1) = 3 and F(1,5,2) = 12 taken as a base, F(1,5,n) = n(n+1)(n+2)(n+3)/24 + F(1,3,n) + F(1,5,n-2). [F(1,3,n) is the number of bracelets with 1 blue, 3 identical red and n identical black beads. If F(1,3,1) = 2 and F(1,3,2) = 6 taken as a base F(1,3,n) = n(n+1)/2 + [|n/2|] + 1 + F(1,3,n-2)], where [|x|]: if a is an integer and a<=x<a+1, [|x|]=a. - Ata A. Uslu and Hamdi G. Ozmenekse, Mar 16 2012

REFERENCES

S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

L. Smith, Polynomial invariants of finite groups, Bull. Am. Math. Soc. 34 (1997), 211-250.

Winston C. Yang (paper in preparation).

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

N. J. A. Sloane, Classic Sequences

Ata A. Uslu and Hamdi G. Ozmenekse, F(1,3,n)

Ata A. Uslu and Hamdi G. Ozmenekse, F(1,5,n)

FORMULA

G.f.: (1+3*x^2)/((1-x)^3*(1-x^2)^3).

a(2n-1) = n(n+1)(n+2)(2n+1)(2n+3)/30, a(2n) = (n+1)(n+2)(n+3)(4n^2+6n+5)/ 30. [M. F. Hasler, May 01 2009]

a(n) = (1/240)*(n+1)*(n+2)*(n+3)*(n+4)*(n+5) + (1/16)*(n+2)*(n+4)*(1/2)*(1+(-1)^n). [Yosu Yurramendi, Jun 20 2013]

MAPLE

a:= n-> (Matrix([[1, 0$6, -3, -9]]). Matrix(9, (i, j)-> if (i=j-1) then 1 elif j=1 then [3, 0, -8, 6, 6, -8, 0, 3, -1][i] else 0 fi)^n)[1, 1]: seq(a(n), n=0..40); # Alois P. Heinz, Jul 31 2008

MATHEMATICA

a[n_?OddQ] := 1/240*(n+1)*(n+2)*(n+3)*(n+4)*(n+5); a[n_?EvenQ] := 1/240*(n+2)*(n+4)*(n+6)*(n^2+3*n+5); Table[a[n], {n, 0, 32}] (* Jean-Fran├žois Alcover, Mar 17 2014, after M. F. Hasler *)

PROG

(PARI) a(k)= if(k%2, (k+1)*(k+3)*(k+5), (k+6)*(k^2+3*k+5))*(k+2)*(k+4)/240 \\ M. F. Hasler, May 01 2009

CROSSREFS

Cf. A160770.

Sequence in context: A066643 A140065 A115549 * A034503 A026557 A124052

Adjacent sequences:  A005992 A005993 A005994 * A005996 A005997 A005998

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Winston C. Yang (yang(AT)math.wisc.edu)

STATUS

approved

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Last modified August 28 15:16 EDT 2015. Contains 261125 sequences.